Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol98.Issue1.Sep2000:
Journal of Econometrics 98 (2000) 107}127
A test for constant correlations in
a multivariate GARCH model
Y.K. Tse*
Department of Economics, National University of Singapore, Singapore 119260, Singapore
Received 1 April 1998; received in revised form 1 December 1998; accepted 1 October 1999
Abstract
We introduce a Lagrange Multiplier (LM) test for the constant-correlation hypothesis
in a multivariate GARCH model. The test examines the restrictions imposed on a model
which encompasses the constant-correlation multivariate GARCH model. It requires the
estimates of the constant-correlation model only and is computationally convenient. We
report some Monte Carlo results on the "nite-sample properties of the LM statistic. The
LM test is compared against the Information Matrix (IM) test due to Bera and Kim
(1996). The LM test appears to have good power against the alternatives considered and
is more robust to nonnormality. We apply the test to three data sets, namely, spot-futures
prices, foreign exchange rates and stock market returns. The results show that the
spot-futures and foreign exchange data have constant correlations, while the correlations
across national stock market returns are time varying. ( 2000 Elsevier Science S.A. All
rights reserved.
JEL classixcation: C12
Keywords: Constant correlation; Information matrix test; Lagrange multiplier test;
Monte Carlo experiment; Multivariate conditional heteroscedasticity
1. Introduction
The success of the autoregressive conditional heteroscedasticity (ARCH) model
and the generalized ARCH (GARCH) model in capturing the time-varying
* Tel.: #65-7723954; fax: #65-7752646.
E-mail address: [email protected] (Y.K. Tse).
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 8 0 - 9
108
Y.K. Tse / Journal of Econometrics 98 (2000) 107}127
variances of economic data in the univariate case has motivated many researchers to extend these models to the multivariate dimension. There are many
examples in which empirical multivariate models of conditional heteroscedasticity can be used fruitfully. An illustrative list includes the following: model the
changing variance structure in an exchange rate regime (Bollerslev, 1990),
calculate the optimal debt portfolio in multiple currencies (Kroner and Claessens, 1991), evaluate the multiperiod hedge ratios of currency futures (Lien and
Luo, 1994), examine the international transmission of stock returns and volatility (Karolyi, 1995) and estimate the optimal hedge ratio for stock index futures
(Park and Switzer, 1995).
Bollerslev et al. (1988) provided the basic framework for a multivariate
GARCH model. They extended the GARCH representation in the univariate
case to the vectorized conditional-variance matrix. While this so-called vech
representation is very general, empirical applications would require further
restrictions and more speci"c structures. A popular member of the vech-representation family is the diagonal form. Under the diagonal form, each variance}covariance term is postulated to follow a GARCH-type equation with
the lagged variance}covariance term and the product of the corresponding lagged residuals as the right-hand-side variables in the conditional(co)variance equation. An advantage of this formulation is that the intuition of
the GARCH model, which has been found to be very successful, is formally
adhered to.
It is often di$cult to verify the condition that the conditional-variance matrix
of an estimated multivariate GARCH model is positive de"nite. Furthermore,
such conditions are often very di$cult to impose during the optimisation of the
log-likelihood function. However, if we postulate the simple assumption that the
correlations are time invariant, these di$culties nicely disappear. Bollerslev
(1990) pointed out that under the assumption of constant correlations, the
maximum likelihood estimate (MLE) of the correlation matrix is equal to the
sample correlation matrix. When the correlation matrix is concentrated out of
the log-likelihood function further simpli"cation is achieved in the optimisation.
As the sample correlation matrix is always positive de"nite, the optimisation
will not fail as long as the conditional variances are positive.
Recently, Engle and Kroner (1995) proposed a class of multivariate conditional heteroscedasticity models called the BEKK (named after Baba, Engle,
Kraft and Kroner) model. The motivation is to ensure the condition of a positive-de"nite conditional-variance matrix in the process of optimisation. Engle
and Kroner provided some theoretical analysis of the BEKK model and related
it to the vech-representation form. Another approach examines the conditional
variance as a factor model. The works by Diebold and Nerlove (1989), Engel and
Rodrigues (1989) and Engle et al. (1990) were along this line. One disadvantage
of the BEKK and factor models is that the parameters cannot be easily
interpreted, and their net e!ects on the future variances and covariances are not
Y.K. Tse / Journal of Econometrics 98 (2000) 107}127
109
readily seen. In other words, the intuitions of the e!ects of the parameters in
a univariate GARCH equation are lost.
Due to its computational simplicity, the constant-correlation GARCH model
is very popular among empirical researchers. Empirical research that uses this
model includes: Bollerslev (1990), Kroner and Claessens (1991), Kroner and
Sultan (1991, 1993), Park and Switzer (1995) and Lien and Tse (1998). However,
the following problems often seem to be overlooked in empirical applications.
First, the assumption of constant correlation is often taken for granted and
seldom analysed or tested. A notable exception, however, is the work by Bera
and Kim (1996). Bera and Kim suggested an Information Matrix (IM) test for
the constant-correlation hypothesis in a bivariate GARCH model and applied
the test to examine the correlation across national stock markets. Second, the
e!ects of the assumption on the conditional-variance estimates are rarely considered. In other words, Are the estimates of the parameters in the conditionalvariance equations robust with respect to the constant-correlation assumption?
In this paper we focus on the "rst question. Our objective is to provide
a convenient test (without having to estimate an encompassing model) for the
constant-correlation assumption and examine the properties of the test in small
samples.
Bollerslev (1990) suggested some diagnostics for the constant-correlation
multivariate GARCH model. He computed the Ljung}Box portmanteau statistic on the cross products of the standardised residuals across di!erent equations.
Critical values were based on the s2 distribution. Another diagnostic was based
on the regression involving the products of the standardised residuals. It was,
however, pointed out by Li and Mak (1994) that the portmanteau statistic is not
asymptotically a s2 and the use of a s2 approximation is inappropriate. For the
residual-based diagnostics, there are usually no su$cient guidelines as to the
choice of regressors in the arti"cial regression. Furthermore, the optimality of
the portmanteau and residual-based tests is not established.
We propose a test for the constant-correlation hypothesis based on the
Lagrange Multiplier (LM) approach. We extend the constant-correlation model
to one in which the correlations are allowed to be time varying. When certain
key parameters in the extended model are imposed to be zero, the constantcorrelation model is obtained. We consider the LM test for the zero restrictions
on the key parameters. Finite-sample properties of the LM test are examined
using Monte Carlo methods. The LM test is compared against the IM test
due to Bera and Kim (1996). We "nd that the LM test has good approximate nominal size in sample sizes of 1000 or above. It is powerful against
the alternative models with time-varying correlations considered. On the other
hand, while the IM test has good approximate nominal size, it lacks
power. Empirical illustrations using real data, however, show that the IM
test rejects the constant-correlation hypothesis vehemently with very low
p values. To explain this anomaly, we examine the behaviour of the tests under
110
Y.K. Tse / Journal of Econometrics 98 (2000) 107}127
nonnormality. It is found that while the Monte Carlo results show that the IM
test leads to gross over-rejection when the errors are nonnormal, the LM test is
relatively robust against nonnormality.
The plan of the rest of the paper is as follows. In Section 2 we derive the LM
statistic. Some Monte Carlo results on the "nite-sample distributions of the LM
and IM tests are reported in Section 3. Section 4 describes some illustrative
examples using real data. In Section 5 we examine the e!ects of nonnormality on
the tests. Finally, we give some concluding remarks in Section 6.
2. The test statistic
Consider a multivariate time series of observations My N, t"1,2, ¹, with
t
K elements each, so that y "(y ,2, y )@. To focus on the conditional heterot
1t
Kt
scedasticity of the time series, we assume that the observations are of zero (or
known) means. This assumption simpli"es tremendously the discussions without straining the notations.
The conditional variance of y is assumed to follow the time-varying structure
t
given by
Var(y DU )"X ,
t t~1
t
where U is the information set at time t. We denote the variance elements of
t
X by p2 , for i"1,2, K, and the covariance elements by p , where
it
ijt
t
1)i(j)K. Following Bollerslev (1990), we consider the constant-correlation
model in which the conditional variances of y follow a GARCH process, while
it
the correlations are constant. Denoting C"Mo N as the correlation matrix, we
ij
have
p2 "u #a p2 #b y2 ,
i i,t~1
it
i
i i,t~1
p "o p p , 1)i(j)K.
ijt
ij it jt
i"1,2, K
(1)
(2)
We assume that u , a and b are nonnegative, a #b (1, for i"1,2, K and
i i
i
i
i
C is positive de"nite. Although the conditional variances in the above equations
are assumed to follow low-order GARCH(1, 1) processes, the test derived below
can be extended to the general GARCH(p, q) models without di$culties.
As pointed out by Bollerslev (1990), the constant-correlation model is computationally attractive. Speci"cally, the MLE of the correlation matrix is equal
to the sample correlation matrix of the standardised residuals, which is always
positive de"nite. The correlation matrix can be further concentrated out from
the log-likelihood function, resulting in a reduction in the number of parameters
to be optimised. Furthermore, it is relatively easy to control the parameters of
the conditional-variance equations during the optimisation so that p2 are
it
Y.K. Tse / Journal of Econometrics 98 (2000) 107}127
111
always positive. On the other hand, it is very di$cult to control a matrix of
parameters to be positive de"nite during the optimisation.
To test for the validity of the constant-correlation assumption, we extend the
above framework to include time-varying correlations. Under some restrictions
on the parameter values of the extended model the constant-correlation model is
derived. The LM test can then be applied to test for the restrictions. This
approach only requires estimates under the constant-correlation model, and can
thus conveniently exploit the computational simplicity of the model.
To allow for time-varying correlations, we consider the following equations
for the correlations:
o "o #d y
y
,
(3)
ijt
ij
ij i,t~1 j,t~1
where d for 1)i(j)K are additional parameters in the extended model.
ij
Thus, the correlations are assumed to respond to the products of previous
observations. From (3) the conditional covariances are given by
p "o p p .
(4)
ijt
ijt it jt
Note that there are N"K2#2K parameters in the extended model with
time-varying correlations. The constant-correlation hypothesis can be tested by
examining the hypothesis H : d "0, for 1)i(j)K. Under H , there are
0 ij
0
M"K(K!1)/2 independent restrictions.
It should be pointed out that (3) is speci"ed as a convenient alternative that
encompasses the constant-correlation model. To ensure that the alternative
model provides well-de"ned positive-de"nite conditional-variance matrices, further restrictions have to be imposed on the parameters d . As in the case of the
ij
general vech speci"cation, such restrictions are very di$cult to derive. Indeed,
empirical research using the vech speci"cation often leaves the issue as an
empirical problem to be resolved in the optimisation stage. As our interest is in
the model under the null H , we shall not pursue the issue of searching for the
0
necessary restrictions. Thus, we assume that within a neighbourhood of d "0,
ij
the optimal properties of the LM test (such as its asymptotic e$ciency against
local alternatives) hold under some regularity conditions as stated in, for
example, Godfrey (1988).
As correlations are standardised measures, it might be arguable to allow the
correlations to depend on the products of the lagged standardised residuals
instead. Thus, if we de"ne e "y /p as the standardised residual, an alternative
it
it it
model might be written as
e
.
(5)
o "o #d@ e
ij i,t~1 j,t~1
ijt
ij
As e depends on other parameters of the model through p , analytic derivation
it
it
of the LM statistic is intractable. If indeed (5) describes the true model, there
may be some loss in power in using (3) as the alternative hypothesis. In return,
however, we obtain analytical tractability. Of course, there is no apriori reason
112
Y.K. Tse / Journal of Econometrics 98 (2000) 107}127
that (5) would provide a better alternative than (3). Indeed, whether using (3)
as the encompassing model would provide a test with good power is an
empirical question. We shall examine its performance using Monte Carlo
methods. Now we shall proceed to derive the LM statistic of H under the above
0
framework.
We denote D as the diagonal matrix with diagonal elements given by p , and
t
it
C "Mo N as the time-varying correlation matrix. Hence the conditional-varit
ijt
ance matrix of y is given by X "D C D . Under the normality assumption the
t
t
t t t
conditional log-likelihood of the observation at time t is given by (the constant
term is ignored)
1
1
l "! ln DD C D D! y@ D~1C~1D~1y
t t
t
t t t
t
2 t t
2
1
1 K
1
"! ln DC D! + ln p2 ! y@ D~1C~1D~1y ,
t t
t
t
it 2 t t
2
2
i/1
for t"1,2, ¹, and the log-likelihood function l is given by l"+T l . For
t/1 t
simplicity, we have assumed that y and p2 are "xed and known. This
i0
i0
assumption has no e!ects on the asymptotic distributions of the LM statistic.
Note that D~1y represents the standardised observations with unit variance.
t t
We denote D~1y "e "(e ,2,e )@.
t t
t
1t
Kt
We now de"ne the following derivatives of p2 with respect to u , a and b for
it
i i
i
i"1,2, K,
d "Lp2 /Lu ,
it
i
it
e "Lp2 /La , f "Lp2 /Lb .
it i
it i
it
it
To calculate these derivatives, the following recursions may be used:
d "1#a d
,
it
i i,t~1
,
e "p2 #a e
i,t~1
i i,t~1
it
f "a f
#y2 ,
i,t~1
it
i i,t~1
(6)
where the starting values are given by d "1, e "p2 and f "y2 .
i0
i0
i1
i1
i1
The "rst partial derivatives of l with respect to the model parameters are
t
given by
(eHe !1)d
Ll
it ,
t " it it
2p2
Lu
it
i
(eHe !1)e
Ll
it ,
t " it it
2p2
La
it
i
Y.K. Tse / Journal of Econometrics 98 (2000) 107}127
113
Ll
(eHe !1) f
t " it it
it ,
Lb
2p2
i
it
Ll
t "eHeH !oij,
t
it jt
Lo
ij
Ll
t "(eHeH !oij)y
y
,
t i,t~1 j,t~1
it jt
Ld
ij
(7)
where eH"(eH ,2,eH )@"C~1e , and C~1"MoijN. Thus, if we denote the parat
t
t t
Kt
1t
t
meters of the model as h"(u , a , b , u ,2, b , o , o ,2,o
,
1 1 1 2
K 12 13
K~1, K
d ,2,d
)@, we can calculate Ll /Lh from the above equations. These
12
K~1, K
t
analytic derivatives can facilitate the evaluation of the MLE of the extended
model if desired. Note that on H , C "C for all t, so that eH"C~1e and
t
t
0 t
oij"oij. In this case, e are just the standardised residuals calculated from the
t
t
algorithm suggested by Bollerslev (1990). We shall denote hK as the MLE of
h under H .
0
If we denote s as the N-element score vector given by s"Ll/Lh and < as the
N]N information matrix given by
A test for constant correlations in
a multivariate GARCH model
Y.K. Tse*
Department of Economics, National University of Singapore, Singapore 119260, Singapore
Received 1 April 1998; received in revised form 1 December 1998; accepted 1 October 1999
Abstract
We introduce a Lagrange Multiplier (LM) test for the constant-correlation hypothesis
in a multivariate GARCH model. The test examines the restrictions imposed on a model
which encompasses the constant-correlation multivariate GARCH model. It requires the
estimates of the constant-correlation model only and is computationally convenient. We
report some Monte Carlo results on the "nite-sample properties of the LM statistic. The
LM test is compared against the Information Matrix (IM) test due to Bera and Kim
(1996). The LM test appears to have good power against the alternatives considered and
is more robust to nonnormality. We apply the test to three data sets, namely, spot-futures
prices, foreign exchange rates and stock market returns. The results show that the
spot-futures and foreign exchange data have constant correlations, while the correlations
across national stock market returns are time varying. ( 2000 Elsevier Science S.A. All
rights reserved.
JEL classixcation: C12
Keywords: Constant correlation; Information matrix test; Lagrange multiplier test;
Monte Carlo experiment; Multivariate conditional heteroscedasticity
1. Introduction
The success of the autoregressive conditional heteroscedasticity (ARCH) model
and the generalized ARCH (GARCH) model in capturing the time-varying
* Tel.: #65-7723954; fax: #65-7752646.
E-mail address: [email protected] (Y.K. Tse).
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 8 0 - 9
108
Y.K. Tse / Journal of Econometrics 98 (2000) 107}127
variances of economic data in the univariate case has motivated many researchers to extend these models to the multivariate dimension. There are many
examples in which empirical multivariate models of conditional heteroscedasticity can be used fruitfully. An illustrative list includes the following: model the
changing variance structure in an exchange rate regime (Bollerslev, 1990),
calculate the optimal debt portfolio in multiple currencies (Kroner and Claessens, 1991), evaluate the multiperiod hedge ratios of currency futures (Lien and
Luo, 1994), examine the international transmission of stock returns and volatility (Karolyi, 1995) and estimate the optimal hedge ratio for stock index futures
(Park and Switzer, 1995).
Bollerslev et al. (1988) provided the basic framework for a multivariate
GARCH model. They extended the GARCH representation in the univariate
case to the vectorized conditional-variance matrix. While this so-called vech
representation is very general, empirical applications would require further
restrictions and more speci"c structures. A popular member of the vech-representation family is the diagonal form. Under the diagonal form, each variance}covariance term is postulated to follow a GARCH-type equation with
the lagged variance}covariance term and the product of the corresponding lagged residuals as the right-hand-side variables in the conditional(co)variance equation. An advantage of this formulation is that the intuition of
the GARCH model, which has been found to be very successful, is formally
adhered to.
It is often di$cult to verify the condition that the conditional-variance matrix
of an estimated multivariate GARCH model is positive de"nite. Furthermore,
such conditions are often very di$cult to impose during the optimisation of the
log-likelihood function. However, if we postulate the simple assumption that the
correlations are time invariant, these di$culties nicely disappear. Bollerslev
(1990) pointed out that under the assumption of constant correlations, the
maximum likelihood estimate (MLE) of the correlation matrix is equal to the
sample correlation matrix. When the correlation matrix is concentrated out of
the log-likelihood function further simpli"cation is achieved in the optimisation.
As the sample correlation matrix is always positive de"nite, the optimisation
will not fail as long as the conditional variances are positive.
Recently, Engle and Kroner (1995) proposed a class of multivariate conditional heteroscedasticity models called the BEKK (named after Baba, Engle,
Kraft and Kroner) model. The motivation is to ensure the condition of a positive-de"nite conditional-variance matrix in the process of optimisation. Engle
and Kroner provided some theoretical analysis of the BEKK model and related
it to the vech-representation form. Another approach examines the conditional
variance as a factor model. The works by Diebold and Nerlove (1989), Engel and
Rodrigues (1989) and Engle et al. (1990) were along this line. One disadvantage
of the BEKK and factor models is that the parameters cannot be easily
interpreted, and their net e!ects on the future variances and covariances are not
Y.K. Tse / Journal of Econometrics 98 (2000) 107}127
109
readily seen. In other words, the intuitions of the e!ects of the parameters in
a univariate GARCH equation are lost.
Due to its computational simplicity, the constant-correlation GARCH model
is very popular among empirical researchers. Empirical research that uses this
model includes: Bollerslev (1990), Kroner and Claessens (1991), Kroner and
Sultan (1991, 1993), Park and Switzer (1995) and Lien and Tse (1998). However,
the following problems often seem to be overlooked in empirical applications.
First, the assumption of constant correlation is often taken for granted and
seldom analysed or tested. A notable exception, however, is the work by Bera
and Kim (1996). Bera and Kim suggested an Information Matrix (IM) test for
the constant-correlation hypothesis in a bivariate GARCH model and applied
the test to examine the correlation across national stock markets. Second, the
e!ects of the assumption on the conditional-variance estimates are rarely considered. In other words, Are the estimates of the parameters in the conditionalvariance equations robust with respect to the constant-correlation assumption?
In this paper we focus on the "rst question. Our objective is to provide
a convenient test (without having to estimate an encompassing model) for the
constant-correlation assumption and examine the properties of the test in small
samples.
Bollerslev (1990) suggested some diagnostics for the constant-correlation
multivariate GARCH model. He computed the Ljung}Box portmanteau statistic on the cross products of the standardised residuals across di!erent equations.
Critical values were based on the s2 distribution. Another diagnostic was based
on the regression involving the products of the standardised residuals. It was,
however, pointed out by Li and Mak (1994) that the portmanteau statistic is not
asymptotically a s2 and the use of a s2 approximation is inappropriate. For the
residual-based diagnostics, there are usually no su$cient guidelines as to the
choice of regressors in the arti"cial regression. Furthermore, the optimality of
the portmanteau and residual-based tests is not established.
We propose a test for the constant-correlation hypothesis based on the
Lagrange Multiplier (LM) approach. We extend the constant-correlation model
to one in which the correlations are allowed to be time varying. When certain
key parameters in the extended model are imposed to be zero, the constantcorrelation model is obtained. We consider the LM test for the zero restrictions
on the key parameters. Finite-sample properties of the LM test are examined
using Monte Carlo methods. The LM test is compared against the IM test
due to Bera and Kim (1996). We "nd that the LM test has good approximate nominal size in sample sizes of 1000 or above. It is powerful against
the alternative models with time-varying correlations considered. On the other
hand, while the IM test has good approximate nominal size, it lacks
power. Empirical illustrations using real data, however, show that the IM
test rejects the constant-correlation hypothesis vehemently with very low
p values. To explain this anomaly, we examine the behaviour of the tests under
110
Y.K. Tse / Journal of Econometrics 98 (2000) 107}127
nonnormality. It is found that while the Monte Carlo results show that the IM
test leads to gross over-rejection when the errors are nonnormal, the LM test is
relatively robust against nonnormality.
The plan of the rest of the paper is as follows. In Section 2 we derive the LM
statistic. Some Monte Carlo results on the "nite-sample distributions of the LM
and IM tests are reported in Section 3. Section 4 describes some illustrative
examples using real data. In Section 5 we examine the e!ects of nonnormality on
the tests. Finally, we give some concluding remarks in Section 6.
2. The test statistic
Consider a multivariate time series of observations My N, t"1,2, ¹, with
t
K elements each, so that y "(y ,2, y )@. To focus on the conditional heterot
1t
Kt
scedasticity of the time series, we assume that the observations are of zero (or
known) means. This assumption simpli"es tremendously the discussions without straining the notations.
The conditional variance of y is assumed to follow the time-varying structure
t
given by
Var(y DU )"X ,
t t~1
t
where U is the information set at time t. We denote the variance elements of
t
X by p2 , for i"1,2, K, and the covariance elements by p , where
it
ijt
t
1)i(j)K. Following Bollerslev (1990), we consider the constant-correlation
model in which the conditional variances of y follow a GARCH process, while
it
the correlations are constant. Denoting C"Mo N as the correlation matrix, we
ij
have
p2 "u #a p2 #b y2 ,
i i,t~1
it
i
i i,t~1
p "o p p , 1)i(j)K.
ijt
ij it jt
i"1,2, K
(1)
(2)
We assume that u , a and b are nonnegative, a #b (1, for i"1,2, K and
i i
i
i
i
C is positive de"nite. Although the conditional variances in the above equations
are assumed to follow low-order GARCH(1, 1) processes, the test derived below
can be extended to the general GARCH(p, q) models without di$culties.
As pointed out by Bollerslev (1990), the constant-correlation model is computationally attractive. Speci"cally, the MLE of the correlation matrix is equal
to the sample correlation matrix of the standardised residuals, which is always
positive de"nite. The correlation matrix can be further concentrated out from
the log-likelihood function, resulting in a reduction in the number of parameters
to be optimised. Furthermore, it is relatively easy to control the parameters of
the conditional-variance equations during the optimisation so that p2 are
it
Y.K. Tse / Journal of Econometrics 98 (2000) 107}127
111
always positive. On the other hand, it is very di$cult to control a matrix of
parameters to be positive de"nite during the optimisation.
To test for the validity of the constant-correlation assumption, we extend the
above framework to include time-varying correlations. Under some restrictions
on the parameter values of the extended model the constant-correlation model is
derived. The LM test can then be applied to test for the restrictions. This
approach only requires estimates under the constant-correlation model, and can
thus conveniently exploit the computational simplicity of the model.
To allow for time-varying correlations, we consider the following equations
for the correlations:
o "o #d y
y
,
(3)
ijt
ij
ij i,t~1 j,t~1
where d for 1)i(j)K are additional parameters in the extended model.
ij
Thus, the correlations are assumed to respond to the products of previous
observations. From (3) the conditional covariances are given by
p "o p p .
(4)
ijt
ijt it jt
Note that there are N"K2#2K parameters in the extended model with
time-varying correlations. The constant-correlation hypothesis can be tested by
examining the hypothesis H : d "0, for 1)i(j)K. Under H , there are
0 ij
0
M"K(K!1)/2 independent restrictions.
It should be pointed out that (3) is speci"ed as a convenient alternative that
encompasses the constant-correlation model. To ensure that the alternative
model provides well-de"ned positive-de"nite conditional-variance matrices, further restrictions have to be imposed on the parameters d . As in the case of the
ij
general vech speci"cation, such restrictions are very di$cult to derive. Indeed,
empirical research using the vech speci"cation often leaves the issue as an
empirical problem to be resolved in the optimisation stage. As our interest is in
the model under the null H , we shall not pursue the issue of searching for the
0
necessary restrictions. Thus, we assume that within a neighbourhood of d "0,
ij
the optimal properties of the LM test (such as its asymptotic e$ciency against
local alternatives) hold under some regularity conditions as stated in, for
example, Godfrey (1988).
As correlations are standardised measures, it might be arguable to allow the
correlations to depend on the products of the lagged standardised residuals
instead. Thus, if we de"ne e "y /p as the standardised residual, an alternative
it
it it
model might be written as
e
.
(5)
o "o #d@ e
ij i,t~1 j,t~1
ijt
ij
As e depends on other parameters of the model through p , analytic derivation
it
it
of the LM statistic is intractable. If indeed (5) describes the true model, there
may be some loss in power in using (3) as the alternative hypothesis. In return,
however, we obtain analytical tractability. Of course, there is no apriori reason
112
Y.K. Tse / Journal of Econometrics 98 (2000) 107}127
that (5) would provide a better alternative than (3). Indeed, whether using (3)
as the encompassing model would provide a test with good power is an
empirical question. We shall examine its performance using Monte Carlo
methods. Now we shall proceed to derive the LM statistic of H under the above
0
framework.
We denote D as the diagonal matrix with diagonal elements given by p , and
t
it
C "Mo N as the time-varying correlation matrix. Hence the conditional-varit
ijt
ance matrix of y is given by X "D C D . Under the normality assumption the
t
t
t t t
conditional log-likelihood of the observation at time t is given by (the constant
term is ignored)
1
1
l "! ln DD C D D! y@ D~1C~1D~1y
t t
t
t t t
t
2 t t
2
1
1 K
1
"! ln DC D! + ln p2 ! y@ D~1C~1D~1y ,
t t
t
t
it 2 t t
2
2
i/1
for t"1,2, ¹, and the log-likelihood function l is given by l"+T l . For
t/1 t
simplicity, we have assumed that y and p2 are "xed and known. This
i0
i0
assumption has no e!ects on the asymptotic distributions of the LM statistic.
Note that D~1y represents the standardised observations with unit variance.
t t
We denote D~1y "e "(e ,2,e )@.
t t
t
1t
Kt
We now de"ne the following derivatives of p2 with respect to u , a and b for
it
i i
i
i"1,2, K,
d "Lp2 /Lu ,
it
i
it
e "Lp2 /La , f "Lp2 /Lb .
it i
it i
it
it
To calculate these derivatives, the following recursions may be used:
d "1#a d
,
it
i i,t~1
,
e "p2 #a e
i,t~1
i i,t~1
it
f "a f
#y2 ,
i,t~1
it
i i,t~1
(6)
where the starting values are given by d "1, e "p2 and f "y2 .
i0
i0
i1
i1
i1
The "rst partial derivatives of l with respect to the model parameters are
t
given by
(eHe !1)d
Ll
it ,
t " it it
2p2
Lu
it
i
(eHe !1)e
Ll
it ,
t " it it
2p2
La
it
i
Y.K. Tse / Journal of Econometrics 98 (2000) 107}127
113
Ll
(eHe !1) f
t " it it
it ,
Lb
2p2
i
it
Ll
t "eHeH !oij,
t
it jt
Lo
ij
Ll
t "(eHeH !oij)y
y
,
t i,t~1 j,t~1
it jt
Ld
ij
(7)
where eH"(eH ,2,eH )@"C~1e , and C~1"MoijN. Thus, if we denote the parat
t
t t
Kt
1t
t
meters of the model as h"(u , a , b , u ,2, b , o , o ,2,o
,
1 1 1 2
K 12 13
K~1, K
d ,2,d
)@, we can calculate Ll /Lh from the above equations. These
12
K~1, K
t
analytic derivatives can facilitate the evaluation of the MLE of the extended
model if desired. Note that on H , C "C for all t, so that eH"C~1e and
t
t
0 t
oij"oij. In this case, e are just the standardised residuals calculated from the
t
t
algorithm suggested by Bollerslev (1990). We shall denote hK as the MLE of
h under H .
0
If we denote s as the N-element score vector given by s"Ll/Lh and < as the
N]N information matrix given by